Non-Archimedean volumes of metrized nef line bundles

Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.

Harmonic analysis and statistics of the first Galois cohomology group

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $$f\in H^1(K,T)$$ f ∈ H 1 ( K , T ) with discriminant-like invariant $$\text {inv}(f)\le X$$ inv ( f ) ≤ X as $$X\rightarrow \infty $$ X → ∞ . Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle’s conjecture for the first Galois cohomology group.

Extreme growth plasticity in the early branching sauropodomorph Massospondylus carinatus

There is growing evidence of developmental plasticity in early branching dinosaurs and their outgroups. This is reflected in disparate patterns of morphological and histological change during ontogeny. In fossils, only the osteohistological assessment of annual lines of arrested growth (LAGs) can reveal the pace of skeletal growth. Some later branching non-bird dinosaur species appear to have followed an asymptotic growth pattern, with declining growth rates at increasing ontogenetic ages. By contrast, the early branching sauropodomorph Plateosaurus trossingensis appears to have had plastic growth, suggesting that this was the plesiomorphic condition for dinosaurs. The South African sauropodomorph Massospondylus carinatus is an ideal taxon in which to test this because it is known from a comprehensive ontogenetic series, it has recently been stratigraphically and taxonomically revised, and it lived at a time of ecosystem upheaval following the end-Triassic extinction. Here, we report on the results of a femoral osteohistological study of M. carinatus comprising 20 individuals ranging from embryo to skeletally mature. We find major variability in the spacing of the LAGs and infer disparate body masses for M. carinatus individuals at given ontogenetic ages, contradicting previous studies. These findings are consistent with a high degree of growth plasticity in M. carinatus .

Asymptotic Behavior of Boson Regge Trajectories

The asymptotic behavior of boson Regge trajectories is studied. Upper and lower bounds on the asymptotic growth of the trajectories are obtained using the phase representation for the trajectories and a number of physical requirements. It is shown that, within the assumptions made, the asymptotic behavior of the trajectories is a square root.

Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $${{\mathbb {R}}}^2$$ R 2 perpendicular to an external constant magnetic field of strength $$B>0$$ B > 0 . We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $$\mu \ge B$$ μ ≥ B (in suitable physical units). For this (pure) state we define its local entropy $$S(\Lambda )$$ S ( Λ ) associated with a bounded (sub)region $$\Lambda \subset {{\mathbb {R}}}^2$$ Λ ⊂ R 2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $$\Lambda $$ Λ of finite area $$|\Lambda |$$ | Λ | . In this setting we prove that the leading asymptotic growth of $$S(L\Lambda )$$ S ( L Λ ) , as the dimensionless scaling parameter $$L>0$$ L > 0 tends to infinity, has the form $$L\sqrt{B}|\partial \Lambda |$$ L B | ∂ Λ | up to a precisely given (positive multiplicative) coefficient which is independent of $$\Lambda $$ Λ and dependent on B and $$\mu $$ μ only through the integer part of $$(\mu /B-1)/2$$ ( μ / B - 1 ) / 2 . Here we have assumed the boundary curve $$\partial \Lambda $$ ∂ Λ of $$\Lambda $$ Λ to be sufficiently smooth which, in particular, ensures that its arc length $$|\partial \Lambda |$$ | ∂ Λ | is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $$B=0$$ B = 0 , where an additional logarithmic factor $$\ln (L)$$ ln ( L ) is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space $$\text{ L}^2({{\mathbb {R}}}^2)$$ L 2 ( R 2 ) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case $$B=0$$ B = 0 , the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.